Trigonometric symmetric boundary value method for oscillating solutions including the sine-Gordon and Poisson equations

S.N Jator,Carlo Cattani

doi:10.1080/23311835.2016.1271269

S.N Jator, Carlo Cattani

Open Access

https://doi.org/10.1080/23311835.2016.1271269

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Journal: Cogent Mathematics	Publication Date: Dec 29, 2016
Citations: 1	License type: CC BY 4.0

Affiliation: Austin Peay State University

Abstract

We construct a continuous linear multistep method with trigonometric coefficients from which a symmetric main method as well as additional methods are reproduced. The main and additional methods whose coefficients depend on the frequency and step length are then applied as a trigonometric symmetric boundary value method (SBVM) to solve systems of second-order initial and boundary value problems of the form y″=f(x,y) without first reducing the ordinary differential equation into an equivalent first-order system. Moreover, the method is successfully applied to solve hyperbolic and elliptic partial differential equations, such as the sine-Gordon and the Poisson equations. The stability property of the SBVM is discussed and numerical experiments are performed to show the accuracy of the method.

Highlights

In this paper, we consider the given system of second-order initial value problem (IVP)y = f (x, y), y(x0) = y0, y (x0) = y0, x0 ≤ x ≤ xN, (1)where f :R × Rd → Rd, N > 0 is an integer, and d is the dimension of the system
We propose a symmetric boundary value method (SBVM) in which on the sequence of points {xn}, defined by xn = x0 + nh, h > 0, n = 0, 1, ... , N, the 4-step [xn, yn] ↦ [xn+4 = xn + 4h, yn+4] is given by the main method
We propose a trigonometric SBVM whose coefficients depend on the frequency and step length

Summary

Introduction

We consider the given system of second-order initial value problem (IVP). The methods (2), (3), and (4) are combined and applied as boundary value methods (BVM) which discretize (1) and simultaneously solve the resulting system as given in Amodio, Golik, and Mazzia (1995), Amodio and Mazzia (1995), Brugnano and Trigiante (1998), and Ghelardoni and Marzulli (1995). These methods are applied to solve higher order IVPs by first reducing the problem into an equivalent first-order system. The SBVM is applied to solve systems of second-order initial and boundary value problems of the form = f (x, y) without first reducing the ordinary differential equation (ODE) into an equivalent first-order system. NSolve[ ] for linear problems, while nonlinear problems were solved by the Newton’s method enhanced by the feature FindRoot[ ]

Estimating the frequency

Hyperbolic PDEs

Conclusion